1. Geometrical Construction of Supertwistor Theory
Shikoku-Seminar
Developments of Q.F.T. & String Theory
Jul.28 - Aug
Kazuki Hasebe
Takuma National College of Technology
arXiv:

2. Introduction: Twistor Program
Roger Penrose (1967)
Space-time is taken to be a secondary
construction from the more primitive
twistor notions.
From ``The Road to Reality’’
Space-Time Event
Twistor Space
Incidence Relation

3. Incidence Relation Non-local transformation 4D Minkowski-space
Twistor-space
Light
(Null-line)
Projective complex-line
Non-local transformation

4. Massless particle and Twistor
Free particle
Massless particle
Helicity
Pauli-Lubanski spin-vector

5. Hopf Map: Template of Twistor
Heinz Hopf (1931)
Topological map from sphere to sphere in different dimensions.
1st Hopf map
2nd Hopf map
3rd Hopf map

6. 1st Hopf Map
1st Hopf Map
Hopf spinor
Incidence Relation

7. 2nd Hopf Map
2nd Hopf map
2nd Hopf spinor
S.C. Zhang
& J.P. Hu (2001)

8. Direct Relation to Twistor
Incidence Relation
Constraint
is Hermitian (space-time is real)
Null Twistor
Helicity zero

9. Idea of Supertwistor Incidence relation A. Ferber (1978) Super-twistor
Complexified space-time
Fermion coordinates
Incidence relation
Non-Hermitian
Super-twistor
Complex space-time is postulated.
Fermion number can be even or odd integer.

10. The SUSY Hopf Map The SUSY Hopf map C. Bartocci, U. Bruzzo, G. Landi
(1987)

11. Supertwistor Variables
Super Incidence Relation
Not-complexified
: Super-Hermitian
Supertwistor variables
Even number
:null-supertwistor

12. Super Incidence Relation
Minkowski-superspace
Supertwistor-space
Non-local super-transformation

13. Supertwistor action and Quantization
Helicity
should be even integer.
Twistor function
wave-function for mass-less particle

14. Relation to Lowest Landau Level
Dirac monopole
U(1) connection
One-particle action
LLL-limit

15. Analogies between Twistor and LLL
Massless Condition
Enhanced Symmetry
More Fundamental Quantity than Space-Time
Holomorphicity, Incidence Relations
Complex conjugation = Derivative
Noncommutative Geometry

16. Conclusion Geometrical construction of the supertwistor
based on the SUSY Hopf map.
Properties of this construction
1. Space-time is not complexified.
2. Even number of fermionic components
of twistor is automatically incorporated.
Close Analogy between LLL physics
and Twistor
Does it suggest something deeper??